Solve the following inequality: \(x+y^2 +\sqrt{x-y^2-1} \leq 1\).
Is it true that, if \(b>a+c>0\), then the quadratic equation \(ax^2 +bx+c=0\) has two roots?
Suppose that: \[\frac{x+y}{x-y}+\frac{x-y}{x+y} =3.\] Find the value of the following expression: \[\frac{x^2 +y^2}{x^2-y^2} + \frac{x^2 -y^2}{x^2+y^2}.\]
Compute the following: \[\frac{(2001\times 2021 +100)(1991\times 2031 +400)}{2011^4}.\]
After a circus came back from its country-wide tour, relatives of the animal tamer asked him questions about which animals travelled with the circus.
“Where there tigers?”
“Yes, in fact, there were seven times more tigers than non-tigers.”
“What about monkeys?”
“Yes, there were seven times less monkeys than non-monkeys.”
“Where there any lions?”
What is the answer he gave to this last question?
Solve this equation: \[(x+2010)(x+2011)(x+2012)=(x+2011)(x+2012)(x+2013).\]
The graph of the function \(y=kx+b\) is shown on the diagram below. Compare \(|k|\) and \(|b|\).
Compare the numbers: \(A=2011\times 20122012\times 201320132013\) and \(B= 2013\times 20112011 \times 201220122012\).
Some inhabitants of the Island of Multi-coloured Frogs speak only the truth, and the rest always lie. Three islanders said:
Bree: There are no blue frogs on our island.
Kevin: Bree is a liar. She herself is a blue frog!
Clara: Of course, Bree is a liar. But she’s a red frog.
Are there any blue frogs on this island?
In the family of happy gnomes there is a father, a mother and a child. The names of the family members: Alex, Charlie and Jo. At the dinner table two gnomes made two statements.
Charlie said: “Alex and Jo are of different genders. Alex and Charlie are my parents”.
Alex said: “I am Jo’s father. I am the daughter of Charlie”.
Who is who? That is, what is the name of the father, the mother and the child, if it is known that each gnome lied once, and each told the truth once.